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Licensed Unlicensed Requires Authentication Published by De Gruyter December 10, 2011

On certain optimal diffeomorphisms between closed curves

Andrea Cerri and Barbara Di Fabio
From the journal Forum Mathematicum


The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between shape properties of topological spaces, modeled as continuous real-valued functions defined on the spaces themselves. Roughly speaking, the natural pseudo-distance is defined as the infimum of the change of the functions' values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance. Moreover, we show that, under our assumptions, this optimal homeomorphism is actually a diffeomorphism.

Funding source: Austrian Science Fund (FWF)

Award Identifier / Grant number: P20134-N13

The authors wish to thank P. Frosini for suggesting the problem and for his indispensable support and friendship. Moreover, they gratefully acknowledge F. Cagliari for several helpful comments and stimulating conversations. However, the authors are solely responsible for any possible errors.

Received: 2011-5-26
Revised: 2011-9-26
Published Online: 2011-12-10
Published in Print: 2014-11-1

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